发表机构
Ovidius University of Constanta; Universidad Rey Juan Carlos（康斯坦察奥维迪乌斯大学; 胡安卡洛斯一世国王大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分类了具有主导加权吸收项的多孔介质方程的自相似解，证明其无穷远行为唯一，但原点附近存在三类不同行为：正解、死核解和幂律奇性解。
AI 中文摘要
本文对具有主导空间非齐次吸收项的多孔介质方程 $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), \quad N\geq1, $$ 其中指数满足 $1<p<m$ 且 $\sigma>0$，的自相似解进行了分类。考虑形如 $$ u(x,t)=t^{-\alpha}f(|x|t^{\beta}), \quad \alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \quad \beta=\frac{m-p}{\sigma(m-1)+2(p-1)} $$ 的解，证明了所有解的剖面在无穷远处满足渐近行为 $$ \lim\limits_{\xi\to\infty}\xi^{\sigma/(p-1)}f(\xi)=\left(\frac{1}{p-1}\right)^{1/(p-1)}, $$ 但解在原点附近的行为却大不相同：存在唯一解满足 $f(0)>0$, $f'(0)=0$；存在另一个唯一解使得 $f$ 呈现\emph{死核}，即存在某个 $\xi_0>0$ 使得 $f\equiv0$ 对 $\xi\in[0,\xi_0]$ 成立；最后，存在 $K^*\in(0,\infty)$ 使得对任意 $K\in(0,K^*)$，至少存在一个解满足 $$ \lim\limits_{\xi\to0}\xi^{-(\sigma+2)/(m-p)}f(\xi)=K. $$ 一般解的大时间行为将充分利用这三类自相似解，在后续工作中加以研究。
英文摘要
Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\real^N\times(0,\infty), \quad N\geq1, $$ with exponents $1<p<m$ and $σ>0$, are classified. Looking for solutions in the form $$ u(x,t)=t^{-α}f(|x|t^β), \quad α=\frac{σ+2}{σ(m-1)+2(p-1)}, \quad β=\frac{m-p}{σ(m-1)+2(p-1)}, $$ it is shown that all their profiles satisfy the behavior at infinity given by $$ \lim\limits_{ξ\to\infty}ξ^{σ/(p-1)}f(ξ)=\left(\frac{1}{p-1}\right)^{1/(p-1)}, $$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)>0$, $f'(0)=0$, another unique solution such that $f$ presents a \emph{dead-core}; that is, $f\equiv0$ for $ξ\in[0,ξ_0]$ for some $ξ_0>0$, and, finally, there exists $K^*\in(0,\infty)$ such that, for any $K\in(0,K^*)$, there is at least a solution such that $$ \lim\limits_{ξ\to0}ξ^{-(σ+2)/(m-p)}f(ξ)=K. $$ The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.
